Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{-p^2 + 6p - 9}{p^2 - 2p - 3} \times \dfrac{p + 1}{5p - 25} $
First factor out any common factors. $r = \dfrac{-(p^2 - 6p + 9)}{p^2 - 2p - 3} \times \dfrac{p + 1}{5(p - 5)} $ Then factor the quadratic expressions. $r = \dfrac {-(p - 3)(p - 3)} {(p - 3)(p + 1)} \times \dfrac {p + 1} {5(p - 5)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac { -(p - 3)(p - 3) \times (p + 1)} { (p - 3)(p + 1) \times 5(p - 5)} $ $r = \dfrac {-(p - 3)(p - 3)(p + 1)} {5(p - 3)(p + 1)(p - 5)} $ Notice that $(p - 3)$ and $(p + 1)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac {-\cancel{(p - 3)}(p - 3)(p + 1)} {5\cancel{(p - 3)}(p + 1)(p - 5)} $ We are dividing by $p - 3$ , so $p - 3 \neq 0$ Therefore, $p \neq 3$ $r = \dfrac {-\cancel{(p - 3)}(p - 3)\cancel{(p + 1)}} {5\cancel{(p - 3)}\cancel{(p + 1)}(p - 5)} $ We are dividing by $p + 1$ , so $p + 1 \neq 0$ Therefore, $p \neq -1$ $r = \dfrac {-(p - 3)} {5(p - 5)} $ $ r = \dfrac{-(p - 3)}{5(p - 5)}; p \neq 3; p \neq -1 $